DOCSLIB.ORG

Yang-Mills Configurations on Nearly K¨Ahler and G2

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Yang-Mills Configurations on Nearly K¨Ahler and G2 Gottfried Wilhelm Leibniz Universitat¨ Hannover Mathematisch-physikalische Fakultat¨ Institut fur¨ Theoretische Physik Diploma Thesis Yang-Mills configurations on nearly Kahler¨ and G2-manifolds by Irina Bauer Date of birth: November 28th 1985, Tomsk Supervisors: Prof. Dr. Olaf Lechtenfeld and Prof. Dr. Alexander D. Popov 1st referee: Prof. Dr. Olaf Lechtenfeld 2nd referee: Prof. Dr. Norbert Dragon Hannover, Mai 19th 2010 Erklarung¨ Hiermit versichere ich, dass ich die Diplomarbeit selbstandig¨ und lediglich unter Benutzung der angegebenen Quellen und Hilfsmittel verfasst habe. Hannover, 19.05.2010 Acknowledgements In the first place I thank my advisor Prof. Dr. Olaf Lechtenfeld for giving me the opportunity to be a diploma student in his group, for continuous support from his side during the whole period of the past twelve months and friendly atmosphere. Of course I also thank the Institut fur¨ Theoretische Physik for offering the diploma program. A special thanks goes to my co-advisor, Prof. Dr. Alexander D. Popov, who is most responsible for helping me to organize the entire process of writing of this thesis and to complete it in an adequately way as well as for giving me an outlook of needed mathematical basics and for answering different questions concerning them. He taught me how to write academic papers, spend a lot of time correcting my drafts of the thesis and supported me by discussing with me different topics and problems arising while writing the thesis. I am also thankful to Felix Lubbe for discussing and comparing several calculations and results. Moreover, I also want to thank Dr. Tatiana A. Ivanova for carefully analyzing and checking my drafts for mistakes and giving useful remarks. To the whole group consisting of persons just mentioned I want to express gratitude for a mutual support and a good teamwork. Finally I thank my family, which always supported me during my study of physics and encouraged me in pursuing my interests. i Contents Contents iii 1 Introduction1 2 Mathematical preliminaries7 2.1 Metric structure on vector spaces................... 7 2.1.1 Vector space Rn ......................... 7 2.1.2 Coordinates on Rn ....................... 8 2.1.3 Linear transformation of Rn .................. 8 2.1.4 Euclidean metric on Rn .................... 9 2.1.5 Metric of signature (p; q) on Rn ................ 9 2.1.6 Groups O(p; q) and SO(p; q) .................. 10 2.1.7 Translations and group IO(p; q)................ 10 2.1.8 Conformal group on Rp;q ................... 11 2.1.9 Rp;q as a topological space................... 11 2.2 Symplectic structures on R2n ..................... 12 2.2.1 Symplectic form ! on the space R2n . 12 2.2.2 Transformation group Sp(2n; R) preserving ! . 12 2.2.3 Degenerate and non-degenerate symplectic forms on R2n . 13 2.3 Complex structure on R2n ....................... 13 2.3.1 Complex coordinates on R2n . 13 2.3.2 A complex structure J on R2n . 14 2.3.3 Subgroup GL(n; C) of GL(2n; R) as a group preserving the complex structure J ....................... 14 2.3.4 Complex structure J on the space R2p;2q and Hermitian metric g of signature (p; q)................... 15 2.3.5 The group U(p; q) preserving J and g . 16 2.3.6 Euclidean case as a subcase.................. 17 2.3.7 Symplectic structure ! on R2n and R2p;2q . 17 2.3.8 U(p; q) as a subgroup of Sp(2n; R)............... 18 iii iv CONTENTS 2.4 Algebraic structures on vector spaces................. 18 2.4.1 General notion of algebra................... 18 2.4.2 Associative and non-associative algebras.......... 20 2.4.3 Algebra of quaternions H ................... 20 2.4.4 The group Sp(1) SU(2) as a group preserving the quater- nion structure.......................... 20 2.4.5 Vector space Hn and the group Sp(n) ............ 23 2.4.6 Metric of signature (p; q) with p + q = n on Hn and the group Sp(p; q).......................... 23 2.4.7 Algebra O of octonions .................... 24 2.4.8 Properties of the octonion structure constants . 26 2.4.9 Group G2 and Spin(7) related with the octonion algebra . 26 2.4.10 Grassmann algebra: definition and main properties . 27 2.4.11 Clifford algebra of the vector space Rp;q of signature (p; q) . 29 2.4.12 Generators of Clifford algebra................. 31 2.4.13 The 8-fold periodicity of Clifford algebra.......... 32 2.4.14 Matrix representations of Clifford algebra over C and R . 34 2.4.15 Spinors (fermions) as representation spaces of Clifford al- gebras .............................. 36 2.4.16 Group Spin(p; q)......................... 38 2.4.17 Supersymmetry and superalgebra.............. 39 2.5 Theory of general Lie algebras..................... 40 2.5.1 Lie algebras........................... 41 2.5.2 Matrix Lie algebras....................... 41 2.5.3 Structure constants, Bianchi identities, Killing-Cartan metric 43 2.5.4 Roots and weights ....................... 44 2.5.5 Some elements of representation theory........... 48 2.5.6 Definitions of affine and Kac-Moody algebras . 48 2.6 Elements of Lie group theory ..................... 51 2.6.1 Matrix Lie groups and their relation with Lie algebras . 51 2.6.2 Generic definition of groups.................. 53 2.6.3 Topological groups and smooth Lie groups......... 53 2.6.4 Left, right and adjoint action of a Lie group G on itself . 54 2.7 Homogeneous spaces.......................... 55 2.7.1 Closed subgroup H of a Lie group G . 55 2.7.2 Homogeneous spaces G=H and H G of a Lie group as n equivalence classes....................... 55 2.7.3 Splitting of the Lie algebra Lie G = Lie H (Lie H) . 56 ⊕ ? 2.7.4 Splitting of commutators ................... 57 2.7.5 The Maurer-Cartan equations................. 58 2.7.6 Metric, Levi-Civita connection, affine connection and torsion 58 CONTENTS v 2.7.7 Kahler¨ coset spaces G=H .................... 61 2.7.8 Nearly Kahler¨ coset spaces in d = 6.............. 62 3 Yang-Mills-theory on nearly K¨ahlermanifolds 65 3.1 Yang-Mills equation on manifold M . 65 3.1.1 Principal fibre bundle ..................... 65 3.1.2 Transition functions ...................... 66 3.1.3 Associated fibre bundle .................... 66 3.1.4 Associated vector bundle ................... 67 3.1.5 Connection ........................... 67 3.1.6 Holonomy group........................ 68 3.1.7 Connection one-form...................... 68 3.1.8 Curvature 2-form........................ 69 3.1.9 Bianchi equation ........................ 70 3.2 Yang-Mills equations on G=H ..................... 71 3.2.1 Coset space G=H ........................ 71 3.2.2 Connection on G=H ....................... 72 3.2.3 Yang-Mills equations...................... 72 3.2.4 Hermitian-Yang-Mills equations............... 73 3.2.5 Some explicit solutions of the Hermitian-Yang-Mills equa- tions ............................... 74 3.3 Manifolds with G2-structure...................... 77 3.3.1 Seven-dimensional manifolds with G2-structure . 77 3.3.2 Cones over nearly Kahler¨ 6-manifolds............ 78 3.3.3 Special G2-manifolds of type R G=H with nearly Kahler¨ × coset spaces G=H ........................ 79 3.4 Yang-Mills on G2-manifolds R G=H . 81 × 3.4.1 Yang-Millsequations on spaces R G=H: G-invariant ansatze¨ × and reductions to kink-type equations............ 81 3.4.2 General solution of the G-invariance condition . 83 3.4.3 Calculation of traces over IAIB . 86 3.4.4 Yang-Mills on R SU(3)=U(1) U(1) ............ 88 × × 3.4.5 Yang-Mills on R Sp(2)=Sp(1) U(1) ............ 90 × × 3.4.6 Action .............................. 94 3.4.7 Action for the case R SU(3)=U(1) U(1).......... 97 × × 3.4.8 Action for the case R Sp(2)=Sp(1) U(1).......... 98 × × 3.4.9 Yang-Mills on R G2=SU(3).................. 99 × 3.4.10 Yang-Mills on R S3 S3 . 100 × × 3.4.11 Solutions on R SU(3)=U(1) U(1) . 100 × × 3.4.12 Solutions on R Sp(2)=Sp(1) U(1) . 105 × × 3.4.13 Instanton-anti-instanton chains and dyons . 106 vi CONTENTS A Appendix 109 A.1 Jacobi identities............................. 109 A.2 Jacobi elliptic functions......................... 109 List of Tables 111 Bibliography 113 Chapter 1 Introduction It was noticed a long time ago that the physical structures of the visible world we are living in can elegantly be described by using a mathematical framework. Es- pecially the Einstein’s vision of space-time as a dynamical object of Riemannian geometry demonstrates how the mathematical understanding of physical pro- cesses helps in logical admission of complicated lows which governs all observed physical phenomena. In generally, it is very exiting to see how just theoretical initially apparently useless mathematical models can once be applied in physics. If we look at the existing physical theories, we see that while the combination of the “Standard Model”, a specific four-dimensional quantum field theory based on Yang-Mills theory, and general relativity makes it difficult to describe some phenomena (for example physics at the energy over the Plank scale M 1019 pl ∼ GeV, “coupling unification” and the “hierarchy problem”), string, superstring and more general M-theory are mathematical models which at present are the best candidates for a unification of quantum theory of all matter and interactions with gravity. Although there is still missing a theory with a geometrical inter- pretation, which one actually would expect from a new concept, they produces various remarkable results at the frontier between mathematics and physics. Historically, it was one of the achievements of the last century attained in accelerator experiments which allowed to unify the three fundamental forces of nature, namely electromagnetism, the weak interaction, and the strong in- teraction in one gauge theory SU(3) SU(2) U(1), showing that quaks and W- × × bosons as well as photons form representations of the gauge groups SU(3) and SU(2) U(1), respectively.
Load more

Recommended publications
  • The Unbroken Spectrum of Frobenius Seaweeds II: Type-B and Type-C
    The unbroken spectrum of Frobenius seaweeds II: type-B and type-C Alex Cameron∗, Vincent E. Coll, Jr.∗∗, Matthew Hyatt†, and Colton Magnant†† July 23, 2019 ∗Department of Mathematics, Lehigh University, Bethlehem, PA, USA: [email protected] ∗∗Department of Mathematics, Lehigh University, Bethlehem, PA, USA: [email protected] †FactSet Research Systems, New York, NY, USA: [email protected] †† Department of Mathematics, Clayton State University, Morrow, GA, USA: [email protected] Abstract Analogous to the Type-An−1 = sl(n) case, we show that if g is a Frobenius seaweed subalgebra of Bn = so(2n +1) or Cn = sp(2n), then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicities have a symmetric distribution. Mathematics Subject Classification 2010 : 17B20, 05E15 Key Words and Phrases: Frobenius Lie algebra, seaweed, biparabolic, principal element, Dynkin diagram, spectrum, regular functional, Weyl group 1 Introduction Notation: All Lie algebras will be finite dimensional over C, and the Lie multiplication will be denoted by [-,-]. The index of a Lie algebra is an important algebraic invariant and, for seaweed algebras, is bounded by the algebra’s rank: ind g ≤ rk g, (see [11]). More formally, the index of a Lie algebra g is given by arXiv:1907.08775v1 [math.RT] 20 Jul 2019 ind g = min dim(ker(BF )), F ∈g∗ where F is a linear form on g, and BF is the associated skew-symmetric bilinear Kirillov form, defined by BF (x,y) = F ([x,y]) for x,y ∈ g. On a given g, index-realizing functionals are called regular and exist in profusion, being dense in both the Zariski and Euclidean topologies of g∗.
    [Show full text]
  • (Bi)Parabolic Subalgebras in the Reductive Lie Algebras. Karin Baur, Anne Moreau
    Quasi-reductive (bi)parabolic subalgebras in the reductive Lie algebras. Karin Baur, Anne Moreau To cite this version: Karin Baur, Anne Moreau. Quasi-reductive (bi)parabolic subalgebras in the reductive Lie algebras.. 2008. hal-00348974v1 HAL Id: hal-00348974 https://hal.archives-ouvertes.fr/hal-00348974v1 Preprint submitted on 22 Dec 2008 (v1), last revised 30 Jun 2010 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. QUASI-REDUCTIVE (BI)PARABOLIC SUBALGEBRAS IN THE REDUCTIVE LIE ALGEBRAS. KARIN BAUR AND ANNE MOREAU Abstract. Let g be a finite dimensional Lie algebra, and z its center. We say that g is quasi- reductive if there is f ∈ g∗ such that g(f)/z is a reductive Lie algebra whose center consists of semisimple elements, where g(f) denotes the stabilizer of f in g for the coadjoint action. If g is reductive, then g is quasi-reductive itself, and also every Borel or Levi subalgebra of g. However the parabolic subalgebras of g are not always quasi-reductive (except in types A or C, see [P03]). Biparabolic (or seaweed) subalgebras are the intersection of two parabolic subalgebras whose sum is g.
    [Show full text]
  • The Derived Algebra of a Stabilizer, Families of Coadjoint Orbits, and Sheets, Arxiv:1202.1135V2 (2012)
    The derived algebra of a stabilizer, families of coadjoint orbits, and sheets Anton Izosimov∗ Abstract Let g be a finite-dimensional real or complex Lie algebra, and let µ ∈ g∗. In the first part of the paper, the relation is discussed between the derived algebra of the stabilizer of µ and the set of coadjoint orbits which have the same dimension as the orbit of µ. In the second part, semisimple Lie algebras are considered, and the relation is discussed between the derived algebra of a centralizer and sheets. 1 Introduction Let g be a finite-dimensional real or complex Lie algebra. The group G acts on the dual space g∗ via the coadjoint action, and g∗ is foliated into the orbits of this action. Consider the union of all orbits ∗ ∗ which have the same codimension k. Denote this union by gk. Each of the sets gk is a quasi-affine ∗ algebraic variety. The study of the varieties gk was initiated by A.Kirillov in connection with the orbit method [6], which relates the unitary dual of G to the set of coadjoint orbits g∗/G. The sets ∗ ∗ gk/G appear in this picture as natural strata of g /G, therefore it is important to understand the ∗ geometry of gk for each k. ∗ So, consider a finite-dimensional real or complex Lie algebra g. Let gµ = {x ∈ g | ad x(µ)=0} be the stabilizer of an element µ ∈ g∗ with respect to the coadjoint representation of g. In the present note, the following simple geometric fact is proved: any element ξ ∈ g∗ which is tangent ∗ ∗ to the variety gk at a point µ ∈ g vanishes on the derived algebra of gµ.
    [Show full text]
  • Periodic Automorphisms of Takiff Algebras, Contractions, and Θ-Groups
    October 15, 2007 PERIODIC AUTOMORPHISMS OF TAKIFF ALGEBRAS, CONTRACTIONS, AND θ-GROUPS DMITRI I. PANYUSHEV INTRODUCTION Let G be a connected reductive algebraic group with Lie algebra g. The ground field | is algebraically closed and of characteristic zero. Fundamental results in invariant theory of the adjoint representation of G are primarily associated with C. Chevalley and B. Kostant. Especially, one should distinguish the ”Chevalley restriction theorem” and seminal article of Kostant [5]. Later, Kostant and Rallis extended these results to the isotropy representa- tion of a symmetric variety [6]. In 1975, E.B. Vinberg came up with the theory of θ-groups. This theory generalises and presents in the most natural form invariant-theoretic results previously known for the adjoint representation and isotropy representations of the sym- metric varieties. Let us remind the main construction and results of Vinberg's article [15]. Let θ Aut(g) 2 be a periodic (= finite order) automorphism of g. The order of θ is denoted by θ . Fix a jθj j j primitive root of unity ζ = p1 and consider the periodic grading (or Zjθj-grading) g = gi ; i Z M2 jθj i θ where gi is the ζ -eigenspace of θ. In particular, g0 = g is the fixed point subalgebra for θ. Let G0 be the connected subgroup of G with Lie algebra g0. The restriction of the adjoint representation yields the natural homomorphism G GL(g ). The linear groups 0 ! 1 obtained in this way are called θ-groups, and the point is that they have the best possible invariant-theoretic properties: |[g ]G0 is a polynomial algebra; • 1 the quotient morphism π : g g ==G = Spec(|[g ]G0 ) is flat; • 1 ! 1 0 1 each fibre of π contains finitely many G -orbits.
    [Show full text]
  • Coadjoint Orbits of Lie Algebras and Cartan Class
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 002, 20 pages Coadjoint Orbits of Lie Algebras and Cartan Class Michel GOZE y and Elisabeth REMM z y Ramm Algebra Center, 4 rue de Cluny, F-68800 Rammersmatt, France E-mail: [email protected] z Universit´ede Haute-Alsace, IRIMAS EA 7499, D´epartement de Math´ematiques, F-68100 Mulhouse, France E-mail: [email protected] Received September 13, 2018, in final form December 31, 2018; Published online January 09, 2019 https://doi.org/10.3842/SIGMA.2019.002 Abstract. We study the coadjoint orbits of a Lie algebra in terms of Cartan class. In fact, the tangent space to a coadjoint orbit O(α) at the point α corresponds to the characteristic space associated to the left invariant form α and its dimension is the even part of the Cartan class of α. We apply this remark to determine Lie algebras such that all the nontrivial orbits (nonreduced to a point) have the same dimension, in particular when this dimension is 2 or 4. We determine also the Lie algebras of dimension 2n or 2n + 1 having an orbit of dimension 2n. Key words: Lie algebras; coadjoint representation; contact forms; Frobenius Lie algebras; Cartan class 2010 Mathematics Subject Classification: 17B20; 17B30; 53D10; 53D05 1 Introduction Let G be a connected Lie group, g its Lie algebra and g∗ the dual vector space of g. We identify g with the Lie algebra of left invariant vector fields on G and g∗ with the vector space of left invariant Pfaffian forms on G.
    [Show full text]
  • Homological Stability of Diffeomorphism Groups 3
    HOMOLOGICAL STABILITY OF DIFFEOMORPHISM GROUPS ALEXANDER BERGLUND AND IB MADSEN Abstract. In this paper we prove a stability theorem for block diffeomor- phisms of 2d-dimensional manifolds that are connected sums of Sd ×Sd. Com- bining this with a recent theorem of S. Galatius and O. Randal-Williams and Morlet’s lemma of disjunction, we determine the homology of the classifying space of their diffeomorphism groups relative to an embedded disk in a stable range. Dedicated to Dennis Sullivan on the occasion of his 70th birthday 1. Introduction The traditional method to obtain homotopical and homological information about diffeomorphism groups of high dimensional smooth manifolds is a two step procedure: the surgery exact sequence handles the larger group of block diffeomor- phisms and Waldhausen’s K-theory of spaces connects block diffeomorphisms and actual diffeomorphisms. In practice one is forced to retreat to rational information. The method was used by Farrell and Hsiang [17] to evaluate the rational homotopy groups of the orientation preserving diffeomorphism groups of spheres in a modest range of dimensions; Q, k ≡ 0 (4), n even n (1) πk(B Diff(S )) ⊗ Q = Q ⊕ Q, k ≡ 0 (4), n odd 0, otherwise. n n for 0 <k< 6 − 7. The orthogonal group SO(n + 1) is a subgroup of Diff(S ), and πkSO(n + 1) accounts for one Q when k ≡ 0 (4). The second Q in line two above is of a different nature. It comes from the connection between Waldhausen’s algebraic K-theory of spaces and the homotopy theory of the homogeneous space Diff(Sn)/ Diff(Sn) of block diffeomorphisms modulo diffeomorphisms.
    [Show full text]
  • Clifford Algebras and the Classical Dynamical Yang-Baxter Equation
    Mathematical Research Letters 10, 253–268 (2003) CLIFFORD ALGEBRAS AND THE CLASSICAL DYNAMICAL YANG-BAXTER EQUATION A. Alekseev and E. Meinrenken Abstract. We describe a relationship of the classical dynamical Yang-Baxter equation with the following elementary problem for Clifford algebras: Given a vector space V with quadratic form QV , how is the exponential of an element in ∧2(V ) under exterior algebra multiplication related to its exponential under Clifford multiplication? 1. Introduction Let g be a real Lie algebra, equipped with a non-degenerate invariant qua- dratic form Q. Let Θ ∈∧3g be the cubic element defined by the quadratic form and the Lie algebra structure. An element r ∈∧2g is called a classical r-matrix for g if it satisfies the (modified) classical Yang-Baxter equation (CYBE) 1 [r, r]g = Θ 2 for some coupling constant ∈ R. Here [r, r]g is defined using the extension of the Lie bracket to the Schouten bracket on the exterior algebra, [·, ·]g : ∧kg × ∧lg →∧k+l−1g. Drinfeld [8] and Semenov-Tian-Shansky [20] gave a geometric interpretation of the CYBE in terms of Poisson-Lie group structures, and a classification of r-matrices for semi-simple Lie algebras and = 0 was obtained by Belavin-Drinfeld [6]. The CYBE admits an important generalization known as the classical dynam- ical Yang-Baxter equation (CDYBE). Let k ⊂ g be a Lie subalgebra. A classical dynamical r-matrix is a k-equivariant (meromorphic)function r : k∗ →∧2g satisfying the (modified)CDYBE [11] ∂r 1 ∧ e + [r, r]g = Θ. ∂µ i 2 i i i ∗ Here ei is a basis on k with dual basis e ∈ k , and µi are the corresponding coordinates on k∗.
    [Show full text]
  • A Mathematical Approach to Wick Rotations
    A mathematical approach to Wick rotations by Christer Helleland Thesis submitted in fulfilment of the requirements for the degree of PHILOSOPHIAE DOCTOR (PhD) ! Faculty of Science and Technology Department of Mathematics and Physics 2019 University of Stavanger NO-4036 Stavanger NORWAY www.uis.no ©2019 Christer Helleland ISBN: Click to enter ISBN. ISSN: Click to enter ISSN. PhD: Thesis UiS No. Click to enter PhD No. Contents List of Tables6 Acknowledgements8 Part 1. Introduction 9 Part 2. Preliminaries 12 1. Cartan involutions of linear Lie groups 12 2. Geometric invariant theory (GIT) 16 Part 3. Wick-rotations and real GIT 22 1. Introduction 22 2. Mathematical Preliminaries 23 2.1. Real form of a complex vector space 23 2.2. Real slices 24 2.3. Compatible real forms 25 3. Holomorphic Riemannian manifolds 27 3.1. Complexification of real manifolds 27 3.2. Complex differential geometry 28 3.3. Real slices from a frame-bundle perspective 29 4. Lie groups 31 4.1. Complex Lie groups and their real forms 31 4.2. Example: Split G2-holonomy manifolds 32 5. A standard Wick-rotation to a real Riemannian space 34 5.1. Minimal vectors and closure of real semi-simple orbits 34 5.2. Compatible triples and intersection of real orbits 36 5.3. The real Riemannian case 37 5.4. The adjoint action of the Lorentz groups O(n − 1; 1) 38 5.5. Uniqueness of real orbits and the class of complex Lie groups 41 6. Applications to the pseudo-Riemannian setting 45 6.1. Pseudo-Riemannian examples 47 Acknowledgements 48 Appendix A.
    [Show full text]
  • On 2-Plectic Lie Groups
    On 2-plectic Lie groups Mohammad Shafiee and Masoud Aminizadeh Abstract. A 2-plectic Lie group is a Lie group endowed with a 2-plectic structure which is left invariant. In this paper we provide some inter- esting examples of 2-plectic Lie groups. Also we study the structure of the set of Hamiltonian covectors and vectors of a 2-plectic Lie algebra. Moreover, the existence of i-isotropic and i-Lagrangian subgroups are in- vestigated. At last we obtain some results about the reduction of some 2-plectic structures. M.S.C. 2010: 53D05, 17B60. Key words: 2-plectic structure; quadratic Lie algebra; Hamiltonian vector. 1 Introduction A k-plectic manifold (M; !) is a smooth manifold M endowed with a (k + 1)-form ! which is closed and nondegenerate in the sense that ιX ! = 0, X 2 TM, implies that X = 0. In this case ! is called a k-plectic structure. These structures are general version of symplectic structures and they naturally appear in the Hamiltonian formulation of classical fields (see [4]and references therein ). Mathematically, in spite of general case, symplectic structures are very interesting and so they have been studied extensively. However, in recent years, 2-plectic structures also have been considered and these structures also are studied extensively ([1], [8]). An important class of 2-plectic manifolds are 2-plectic Lie groups. Similar to symplectic case, a 2-plectic Lie group is defined as follows. Definition 1.1. A 2-plectic Lie group (G; !) is a Lie group G endowed with a 2- plectic structure ! which is left invariant.
    [Show full text]
  • Clifford Algebras and Lie Groups Lecture Notes, University of Toronto
    Clifford algebras and Lie groups Eckhard Meinrenken Lecture Notes, University of Toronto, Fall 2009. Last revision: 12/2011 Contents Chapter 1. Symmetric bilinear forms 7 1. Quadratic vector spaces 7 2. Isotropic subspaces 9 3. Split bilinear forms 10 4. E.Cartan-Dieudonn´e'sTheorem 13 5. Witt's Theorem 16 6. Orthogonal groups for K = R; C 17 7. Lagrangian Grassmannians 23 Chapter 2. Clifford algebras 27 1. Exterior algebras 27 1.1. Definition 27 1.2. Universal property, functoriality 28 2. Clifford algebras 29 2.1. Definition and first properties 30 2.2. Universal property, functoriality 31 2.3. The Clifford algebras Cl(n; m) 32 2.4. The Clifford algebras Cl(n) 33 2.5. Symbol map and quantization map 34 2.6. Transposition 35 2.7. Chirality element 36 2.8. The trace and the super-trace 37 2.9. Extension of the bilinear form 38 2.10. Lie derivatives and contractions 38 2.11. The Lie algebra q(^2(V )) 40 2.12. A formula for the Clifford product 41 3. The Clifford algebra as a quantization of the exterior algebra 42 3.1. Differential operators 42 3.2. Graded Poisson algebras 44 3.3. Graded super Poisson algebras 45 3.4. Poisson structures on ^(V ) 46 Chapter 3. The spin representation 49 1. The Clifford group and the spin group 49 1.1. The Clifford group 49 1.2. The groups Pin(V ) and Spin(V ) 51 2. Clifford modules 54 2.1. Basic constructions 54 2.2. The spinor module SF 56 2.3.
    [Show full text]
  • Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups Tome 64, No 5 (2014), P
    R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER K. H. NEEB Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups Tome 64, no 5 (2014), p. 1823-1892. <http://aif.cedram.org/item?id=AIF_2014__64_5_1823_0> © Association des Annales de l’institut Fourier, 2014, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 64, 5 (2014) 1823-1892 SEMIBOUNDED UNITARY REPRESENTATIONS OF DOUBLE EXTENSIONS OF HILBERT–LOOP GROUPS by K. H. NEEB (*) Abstract. — A unitary representation π of a, possibly infinite dimensional, Lie group G is called semibounded if the corresponding operators idπ(x) from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra g of G. We classify all irreducible semibounded represen- tations of the groups Lbφ(K) which are double extensions of the twisted loop group Lφ(K), where K is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant) and φ is a finite order automorphism of K which leads to one of the 7 irreducible locally affine root systems with their canonical Z-grading.
    [Show full text]
  • Invariant Nijenhuis Tensors and Integrable Geodesic Flows
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 056, 30 pages Invariant Nijenhuis Tensors and Integrable Geodesic Flows Konrad LOMPERT y and Andriy PANASYUK z y Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland E-mail: [email protected] z Faculty of Mathematics and Computer Science, University of Warmia and Mazury, ul. Sloneczna 54, 10-710 Olsztyn, Poland E-mail: [email protected] Received December 19, 2018, in final form August 02, 2019; Published online August 07, 2019 https://doi.org/10.3842/SIGMA.2019.056 Abstract. We study invariant Nijenhuis (1; 1)-tensors on a homogeneous space G=K of a reductive Lie group G from the point of view of integrability of a Hamiltonian system of differential equations with the G-invariant Hamiltonian function on the cotangent bun- ∗ ∗ dle T (G=K). Such a tensor induces an invariant Poisson tensor Π1 on T (G=K), which is Poisson compatible with the canonical Poisson tensor ΠT ∗(G=K). This Poisson pair can be reduced to the space of G-invariant functions on T ∗(G=K) and produces a family of Poisson commuting G-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions of the completeness of this family. As an application we prove Liouville integra- bility in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces G=K of compact Lie groups G for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of G to two subgroups G = G1 · G2, where G=Gi are symmetric spaces, K = G1 \ G2.
    [Show full text]